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doi: 10.3934/dcdsb.2022099
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## Delay-dependent flocking dynamics of a two-group coupling system

 College of Liberal Arts and Sciences, National University of Defense Technology, Changsha, 410073, China

*Corresponding author: Yicheng Liu

Received  September 2021 Revised  April 2022 Early access June 2022

Fund Project: The first author is supported by the National Natural Science Foundation of China grant 11671011, Postgraduate Scientific Research Innovation Project of Hunan Province grant CX20200011

A group coupling model for a system with large-scale nodes is investigated. The model is formulated as a system of functional differential equations. It incorporates two additional factors that exist in the evolution of flocking behavior, but are often ignored in modeling: (ⅰ) the diversity of interactions, including inter-group and intra-group interactions and (ⅱ) the delayed response of particles to signals from the environment or neighbors, including transmission and processing delays. Theoretically, using the divide-and-conquer method and under different delay factors, sufficient conditions for self-organizing flocking are derived by constructing a dissipative differential inequalities with continuous parameters respectively, which involve some analytical expressions of the upper bound of the delay that the system can tolerate. Results of systematic numerical simulations are presented. They not only validate the analytical results, but hint at a somehow surprising behavior of system, that is, weak flocking behavior occurs when two types of delays coexist.

Citation: Maoli Chen, Yicheng Liu, Xiao Wang. Delay-dependent flocking dynamics of a two-group coupling system. Discrete and Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2022099
##### References:

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##### References:
Schematic diagram of a two-group coupling system subject to processing delay $\tau_1$ and transmission delay $\tau_2$
Schematic diagram of flocking results under Scenario 1
Schematic diagram of particle velocity evolution in Scenario 1 with large time delay
Schematic diagram of flocking results under Scenario 2
Schematic diagram of particle velocity evolution in Scenario 2 with large time delay
Schematic diagram of the relationship between the upper bounds of the two types of time delay and the system parameters. To ensure the flocking behavior, the upper bound of the transmission delay allowed by the system decreases with the increase of the position diameter margin $d$, and the upper bound of the processing delay allowed by the system decreases with the increase of the total number of members $N$
When the transmission delay and the processing delay coexist, a weak flocking phenomenon occurs in system (2)-(3)
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